Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2121 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42×C5 · Schur multiplier (H2) = C22×C4 |
| direct product | C5×#12624 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C5 | C2×#99 |
| commutator | C23 | C2×C42×C5 |
| Frattini, cosocle | C2×#41 | C23×C5 |
| socle | C2×C5 | C4×#99 |
| prime radical | C5×#525 | C22 |
| LCS | G ▷ C23 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C5 ◁ C2×C5×C8 ◁ C2×C5×#131 ◁ G | |
456 conjugacy classes of subgroups.
| minimal degree | μ(G) = 37 |
| transitive degree | μt(G) = 160 |
A minimal faithful permutation representation — 3 generators on 37 points:
(1,2,3,4,5)(6,7,11,9,12,15,26,22)(8,16,19,31,20,29,33,34)(10,14,24,21,25,30,37,35)(13,27,17,23,28,36,32,18);(6,8)(7,13)(9,17)(10,23,25,18)(11,19)(12,20)(14,29,30,16)(15,28)(21,34,35,31)(22,32)(24,36,37,27)(26,33);(6,9,11,15,12,22,26,7)(8,17,19,28,20,32,33,13)(10,21,24,30,25,35,37,14)(16,18,31,27,29,23,34,36)230 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 5612);